Difference between revisions of "2008 IMO Problems/Problem 2"
(New page: == Problem 2 == '''(i)''' If <math>x</math>, <math>y</math> and <math>z</math> are three real numbers, all different from <math>1</math>, such that <math>xyz = 1</math>, then prove that <m...) |
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Revision as of 21:18, 4 September 2008
Problem 2
(i) If ,
and
are three real numbers, all different from
, such that
, then prove that
.
(With the
sign for cyclic summation, this inequality could be rewritten as
.)
(ii) Prove that equality is achieved for infinitely many triples of rational numbers ,
and
.
Solution
Consider the transormation defined by
and put
. Since
maps rational numbers to rational, the problem is equivalent to showing that
given that
\[\frac{\alpha}{\alpha+1)\frac{\beta}{\beta+1) \frac{\gamma}{\gamma+1) = 1 \quad (2)\] (Error compiling LaTeX. Unknown error_msg)
and that the equallity holds for infinitely many triplets of .